The fact that PID loops work for many systems is a consequence of the dynamics of those systems, for example that the integral of velocity is position and the derivative of velocity is acceleration. A vast array of phenomena can be modeled as 2nd order linear ODEs.

Which just means that for many things your brain doesn't need a new model, just to parametrize one of the most common models.

One way to look at a PID controller is that it uses a simple model that is correct in a lot of situations.

Yeah tuning the PID controller is in fact modeling the dynamics of the system. You can't really expect an untuned PID controller to just work.

As a bad physicist ("physics is the art of approximation", some prof said during my studies): Indeed, you only have to model the relevant aspect(s) of the system, which may be completely unrecognizable as the system.

I guess it's somewhat more true in programming, because if you have only ten lines, can you even call it modeling?

That said, I like to say that the best kind of code teaches you something new abou the problem it's solving. Some invariant, widely usable simplification, etc. Still not the same as modeling the system as such, though.

>because if you have only ten lines, can you even call it modeling?

Ten? Do you even Perl?

I think this Wiki page discusses it: https://en.wikipedia.org/wiki/Internal_model_(motor_control)

Yes. That sentence is generally untrue if the context is not specified.

Either the authors made rudimentary error or they specified the context. If you read forward, they define the problem and discuss error and cause controlled regulation.

linear model